Rod arrangements or lever arrangements are usually used, particularly for the automatic production and separation of plug-in connections, in order to shift a plug linearly in the plug-in direction. Mechanical arrangements that are suitable for this purpose require a large structured space in the plug-in direction. Because the plug, in particular together with its plug head, has to protrude completely from the mechanism in order to be inserted correctly into a corresponding plug receptacle, such as, for example, a plug socket, it is necessary to incorporate the fastening of the mechanism at the plug as well as the application of force by way of the mechanism into the plug below the plug head. Accordingly, the structural height of the mechanism overall—including the plug—is greater than the plug height measured in the plug-in direction. This is a drawback, in particular, when the coupling device is intended to be suitable for producing a plug-in connection in the area of the underbody of a motor vehicle and to be capable of being shifted for this purpose under the motor vehicle. Naturally, such an application results in a very restrictive height limitation for the coupling device and hence, at the same time, also for the shifting mechanism of the plug as well as for the plug itself.